I am a newbie to Julia and still trying to figure out everything.
I want to restrict input variable type to array that can contain int and floats.
I would really appreciate any help.
function foo(array::??)
As I mentioned in the comment, you don't want to mix them for performance reasons. However, if your array can be either Floats or Ints, but you don't know which it will be, then the best approach is to make it dispatch on the parametric type:
function foo{T<:Number,N}(array::Array{T,N})
This will make it compile a separate function for arrays of each number type (only when needed), and since the type will be known for the compiler, it will run an optimized version of the function whether you give it foo([0.1,0.3,0.4]), foo([1 2 3]), foo([1//2 3//4]), etc.
Updated syntax in Julia 0.6+
function foo(array::Array{T,N}) where {T<:Number,N}
For more generality, you can use Array{Union{Int64,Float64},N} as a type. This will allow Floats and Ints, and you can use its constructor like
arr = Array{Union{Int64,Float64},2}(4,4) # The 2 is the dimension, (4,4) is the size
and you can allow dispatching onto weird things like this as well by doing
function foo{T,N}(array::Array{T,N})
i.e. just remove the restriction on T. However, since the compiler cannot know in advance whether any element of the array is an Int or a Float, it cannot optimize it very well. So in general you should not do this...
But let me explain one way you can work with this and still get something with decent performance. It also works by multiple dispatch. Essentially, if you encase your inner loops with a function call which is a strictly typed dispatch, then when doing all of the hard calculations it can know exactly what type it is and optimize the code anyways. This is best explained by an example. Let's say we want to do:
function foo{T,N}(array::Array{T,N})
for i in eachindex(array)
val = array[i]
# do algorithm X on val
end
end
You can check using #code_warntype that val will not compile as an Int64 or Float64 because it won't know until runtime what type it will be for each i. If you check #code_llvm (or #code_native for the assembly) you see that there is a really long code that is generated in order to handle this. What we can instead do is define
function inner_foo{T<:Number}(val::T)
# Do algorithm X on val
end
and then instead define foo as
function foo2{T,N}(array::Array{T,N})
for i in eachindex(array)
inner_foo(array[i])
end
end
While this looks the same to you, it is very different to the compiler. Note that inner_foo(array[i]) dispatches a specially-compiled function for whatever number type it sees, so in foo2 algorithm X is calculated efficiently, and the only non-efficient part is the wrapping above inner_foo (so if all your time is spent in inner_foo, you will get basically maximal performance).
This is why Julia is built around multiple-dispatch: it's a design which allows you to push things out to optimized functions whenever possible. Julia is fast because of it. Use it.
This should be a comment to Chris' answer, but I don't have enough points to comment.
As Chris points out, using function barriers can be quite useful to generate optimal code. However be aware that dynamic dispatch has some overhead. This may or may not be important depending on the complexity of the inner function.
function foo1{T,N}(array::Array{T,N})
s = 0.0
for i in eachindex(array)
val = array[i]
s += val*val
end
s
end
function foo2{T,N}(array::Array{T,N})
s = 0.0
for i in eachindex(array)
s += inner_foo(array[i])
end
s
end
function foo3{T,N}(array::Array{T,N})
s = 0.0
for i in eachindex(array)
val = array[i]
if isa(val, Float64)
s += inner_foo(val::Float64)
else
s += inner_foo(val::Int64)
end
end
s
end
function inner_foo{T<:Number}(val::T)
val*val
end
For A = Array{Union{Int64,Float64},N}, foo2 doesn't provide much speedup over foo1 since benefit of the optimised inner_foo is countered by the cost of dynamic dispatch.
foo3 is much faster (~7 times) and could be used if possible types are limited and known ahead of time (as in above example where elements are either Int64 or Float64)
See https://groups.google.com/forum/#!topic/julia-users/OBs0fmNmjCU for further discussion.
To better understand recursion, I'm trying to count how many characters are between each pair of (),
not counting characters that are within other ()s. For example:
(abc(ab(abc)cd)(()ab))
would output:
Level 3: 3
Level 2: 4
Level 3: 0
Level 2: 2
Level 1: 3
Where "Level" refers to the level of () nesting. So level three would mean that the characters are within a pair(1) within a pair(2) within a pair(3).
To do this, my guess is that the easiest thing to do is to implement some sort of recursive call to the function, as commented inside the function "recursiveParaCheck". What is my approach as I begin thinking about a recurrence relationship?
#include <stdio.h>
#include <stdlib.h>
#include <string.h>
#include <ctype.h>
int recursiveParaCheck(char input[], int startPos, int level);
void main()
{
char input[] = "";
char notDone = 'Y';
do
{
//Read in input
printf("Please enter input: ");
scanf(" %s", input);
//Call Recursive Function to print out desired information
recursiveParaCheck(input, 1, 1);
printf("\n Would you like to try again? Y/N: ");
scanf(" %c", ¬Done);
notDone = toupper(notDone);
}while(notDone == 'Y');
}
int recursiveParaCheck(char input[], int startPos, int level)
{
int pos = startPos;
int total = 0;
do
{
if(input[pos] != '(' && input[pos] != ')')
{
++total;
}
//What is the base case?
if(BASE CASE)
{
//Do something?
}
//When do I need to make a recursive call?
if(SITUATION WHERE I MAKE RECURSIVE CALL)
{
//Do something?
}
++pos;
}while(pos < 1000000); // assuming my input will not be this long
}
Recursion is a wonderful programming tool. It provides a simple, powerful way of approaching a variety of problems. It is often hard, however, to see how a problem can be approached recursively; it can be hard to "think" recursively. It is also easy to write a recursive program that either takes too long to run or doesn't properly terminate at all. In this article we'll go over the basics of recursion and hopefully help you develop, or refine, a very important programming skill.
What is Recursion?
In order to say exactly what recursion is, we first have to answer "What is recursion?" Basically, a function is said to be recursive if it calls itself.
You may be thinking this is not terribly exciting, but this function demonstrates some key considerations in designing a recursive algorithm:
It handles a simple "base case" without using recursion.
In this example, the base case is "HelloWorld(0)"; if the function is asked to print zero times then it returns without spawning any more "HelloWorld"s.
It avoids cycles.
Why use Recursion?
The problem we illustrated above is simple, and the solution we wrote works, but we probably would have been better off just using a loop instead of bothering with recursion. Where recursion tends to shine is in situations where the problem is a little more complex. Recursion can be applied to pretty much any problem, but there are certain scenarios for which you'll find it's particularly helpful. In the remainder of this article we'll discuss a few of these scenarios and, along the way, we'll discuss a few more core ideas to keep in mind when using recursion.
Scenario #1: Hierarchies, Networks, or Graphs
In algorithm discussion, when we talk about a graph we're generally not talking about a chart showing the relationship between variables (like your TopCoder ratings graph, which shows the relationship between time and your rating). Rather, we're usually talking about a network of things, people, or concepts that are connected to each other in various ways. For example, a road map could be thought of as a graph that shows cities and how they're connected by roads. Graphs can be large, complex, and awkward to deal with programatically. They're also very common in algorithm theory and algorithm competitions. Luckily, working with graphs can be made much simpler using recursion. One common type of a graph is a hierarchy, an example of which is a business's organization chart:
Name Manager
Betty Sam
Bob Sally
Dilbert Nathan
Joseph Sally
Nathan Veronica
Sally Veronica
Sam Joseph
Susan Bob
Veronica
In this graph, the objects are people, and the connections in the graph show who reports to whom in the company. An upward line on our graph says that the person lower on the graph reports to the person above them. To the right we see how this structure could be represented in a database. For each employee we record their name and the name of their manager (and from this information we could rebuild the whole hierarchy if required - do you see how?).
Now suppose we are given the task of writing a function that looks like "countEmployeesUnder(employeeName)". This function is intended to tell us how many employees report (directly or indirectly) to the person named by employeeName. For example, suppose we're calling "countEmployeesUnder('Sally')" to find out how many employees report to Sally.
To start off, it's simple enough to count how many people work directly under her. To do this, we loop through each database record, and for each employee whose manager is Sally we increment a counter variable. Implementing this approach, our function would return a count of 2: Bob and Joseph. This is a start, but we also want to count people like Susan or Betty who are lower in the hierarchy but report to Sally indirectly. This is awkward because when looking at the individual record for Susan, for example, it's not immediately clear how Sally is involved.
A good solution, as you might have guessed, is to use recursion. For example, when we encounter Bob's record in the database we don't just increment the counter by one. Instead, we increment by one (to count Bob) and then increment it by the number of people who report to Bob. How do we find out how many people report to Bob? We use a recursive call to the function we're writing: "countEmployeesUnder('Bob')". Here's pseudocode for this approach:
function countEmployeesUnder(employeeName)
{
declare variable counter
counter = 0
for each person in employeeDatabase
{
if(person.manager == employeeName)
{
counter = counter + 1
counter = counter + countEmployeesUnder(person.name)
}
}
return counter
}
If that's not terribly clear, your best bet is to try following it through line-by-line a few times mentally. Remember that each time you make a recursive call, you get a new copy of all your local variables. This means that there will be a separate copy of counter for each call. If that wasn't the case, we'd really mess things up when we set counter to zero at the beginning of the function. As an exercise, consider how we could change the function to increment a global variable instead. Hint: if we were incrementing a global variable, our function wouldn't need to return a value.
Mission Statements
A very important thing to consider when writing a recursive algorithm is to have a clear idea of our function's "mission statement." For example, in this case I've assumed that a person shouldn't be counted as reporting to him or herself. This means "countEmployeesUnder('Betty')" will return zero. Our function's mission statment might thus be "Return the count of people who report, directly or indirectly, to the person named in employeeName - not including the person named employeeName."
Let's think through what would have to change in order to make it so a person did count as reporting to him or herself. First off, we'd need to make it so that if there are no people who report to someone we return one instead of zero. This is simple -- we just change the line "counter = 0" to "counter = 1" at the beginning of the function. This makes sense, as our function has to return a value 1 higher than it did before. A call to "countEmployeesUnder('Betty')" will now return 1.
However, we have to be very careful here. We've changed our function's mission statement, and when working with recursion that means taking a close look at how we're using the call recursively. For example, "countEmployeesUnder('Sam')" would now give an incorrect answer of 3. To see why, follow through the code: First, we'll count Sam as 1 by setting counter to 1. Then when we encounter Betty we'll count her as 1. Then we'll count the employees who report to Betty -- and that will return 1 now as well.
It's clear we're double counting Betty; our function's "mission statement" no longer matches how we're using it. We need to get rid of the line "counter = counter + 1", recognizing that the recursive call will now count Betty as "someone who reports to Betty" (and thus we don't need to count her before the recursive call).
As our functions get more and more complex, problems with ambiguous "mission statements" become more and more apparent. In order to make recursion work, we must have a very clear specification of what each function call is doing or else we can end up with some very difficult to debug errors. Even if time is tight it's often worth starting out by writing a comment detailing exactly what the function is supposed to do. Having a clear "mission statement" means that we can be confident our recursive calls will behave as we expect and the whole picture will come together correctly.
Generally, I have a headache because something is wrong with my reasoning:
For 1 set of arguments, referential transparent function will always return 1 set of output values.
that means that such function could be represented as a truth table (a table where 1 set of output parameters is specified for 1 set of arguments).
that makes the logic behind such functions is combinational (as opposed to sequential)
that means that with pure functional language (that has only rt functions) it is possible to describe only combinational logic.
The last statement is derived from this reasoning, but it's obviously false; that means there is an error in reasoning. [question: where is error in this reasoning?]
UPD2. You, guys, are saying lots of interesting stuff, but not answering my question. I defined it more explicitly now. Sorry for messing up with question definition!
Question: where is error in this reasoning?
A referentially transparent function might require an infinite truth table to represent its behavior. You will be hard pressed to design an infinite circuit in combinatory logic.
Another error: the behavior of sequential logic can be represented purely functionally as a function from states to states. The fact that in the implementation these states occur sequentially in time does not prevent one from defining a purely referentially transparent function which describes how state evolves over time.
Edit: Although I apparently missed the bullseye on the actual question, I think my answer is pretty good, so I'm keeping it :-) (see below).
I guess a more concise way to phrase the question might be: can a purely functional language compute anything an imperative one can?
First of all, suppose you took an imperative language like C and made it so you can't alter variables after defining them. E.g.:
int i;
for (i = 0; // okay, that's one assignment
i < 10; // just looking, that's all
i++) // BUZZZ! Sorry, can't do that!
Well, there goes your for loop. Do we get to keep our while loop?
while (i < 10)
Sure, but it's not very useful. i can't change, so it's either going to run forever or not run at all.
How about recursion? Yes, you get to keep recursion, and it's still plenty useful:
int sum(int *items, unsigned int count)
{
if (count) {
// count the first item and sum the rest
return *items + sum(items + 1, count - 1);
} else {
// no items
return 0;
}
}
Now, with functions, we don't alter state, but variables can, well, vary. Once a variable passes into our function, it's locked in. However, we can call the function again (recursion), and it's like getting a brand new set of variables (the old ones stay the same). Although there are multiple instances of items and count, sum((int[]){1,2,3}, 3) will always evaluate to 6, so you can replace that expression with 6 if you like.
Can we still do anything we want? I'm not 100% sure, but I think the answer is "yes". You certainly can if you have closures, though.
You have it right. The idea is, once a variable is defined, it can't be redefined. A referentially transparent expression, given the same variables, always yields the same result value.
I recommend looking into Haskell, a purely functional language. Haskell doesn't have an "assignment" operator, strictly speaking. For instance:
my_sum numbers = ??? where
i = 0
total = 0
Here, you can't write a "for loop" that increments i and total as it goes along. All is not lost, though. Just use recursion to keep getting new is and totals:
my_sum numbers = f 0 0 where
f i total =
if i < length numbers
then f i' total'
else total
where
i' = i+1
total' = total + (numbers !! i)
(Note that this is a stupid way to sum a list in Haskell, but it demonstrates a method of coping with single assignment.)
Now, consider this highly imperative-looking code:
main = do
a <- readLn
b <- readLn
print (a + b)
It's actually syntactic sugar for:
main =
readLn >>= (\a ->
readLn >>= (\b ->
print (a + b)))
The idea is, instead of main being a function consisting of a list of statements, main is an IO action that Haskell executes, and actions are defined and chained together with bind operations. Also, an action that does nothing, yielding an arbitrary value, can be defined with the return function.
Note that bind and return aren't specific to actions. They can be used with any type that calls itself a Monad to do all sorts of funky things.
To clarify, consider readLn. readLn is an action that, if executed, would read a line from standard input and yield its parsed value. To do something with that value, we can't store it in a variable because that would violate referential transparency:
a = readLn
If this were allowed, a's value would depend on the world and would be different every time we called readLn, meaning readLn wouldn't be referentially transparent.
Instead, we bind the readLn action to a function that deals with the action, yielding a new action, like so:
readLn >>= (\x -> print (x + 1))
The result of this expression is an action value. If Haskell got off the couch and performed this action, it would read an integer, increment it, and print it. By binding the result of an action to a function that does something with the result, we get to keep referential transparency while playing around in the world of state.
As far as I understand it, referential transparency just means: A given function will always yield the same result when invoked with the same arguments. So, the mathematical functions you learned about in school are referentially transparent.
A language you could check out in order to learn how things are done in a purely functional language would be Haskell. There are ways to use "updateable storage possibilities" like the Reader Monad, and the State Monad for example. If you're interested in purely functional data structures, Okasaki might be a good read.
And yes, you're right: Order of evaluation in a purely functional language like haskell does not matter as in non-functional languages, because if there are no side effects, there is no reason to do someting before/after something else -- unless the input of one depends on the output of the other, or means like monads come into play.
I don't really know about the truth-table question.
Here's my stab at answering the question:
Any system can be described as a combinatorial function, large or small.
There's nothing wrong with the reasoning that pure functions can only deal with combinatorial logic -- it's true, just that functional languages hide that from you to some extent or another.
You could even describe, say, the workings of a game engine as a truth table or a combinatorial function.
You might have a deterministic function that takes in "the current state of the entire game" as the RAM occupied by the game engine and the keyboard input, and returns "the state of the game one frame later". The return value would be determined by the combinations of the bits in the input.
Of course, in any meaningful and sane function, the input is parsed down to blocks of integers, decimals and booleans, but the combinations of the bits in those values is still determining the output of your function.
Keep in mind also that basic digital logic can be described in truth tables. The only reason that that's not done for anything more than, say, arithmetic on 4-bit integers, is because the size of the truth table grows exponentially.
The error in Your reasoning is the following:
"that means that such function could be represented as a truth table".
You conclude that from a functional language's property of referential transparency. So far the conclusion would sound plausible, but You oversee that a function is able to accept collections as input and process them in contrast to the fixed inputs of a logic gate.
Therefore a function does not equal a logic gate but rather a construction plan of such a logic gate depending on the actual (at runtime determined) input!
To comment on Your comment: Functional languages can - although stateless - implement a state machine by constructing the states from scratch each time they are being accessed.
Closures are poor man's objects and vice versa.
I have seen this statement at many places on the web (including SO) but I don't quite understand what it means. Could someone please explain what it exactly means?
If possible, please include examples in your answer.
Objects are poor man's closures.
Consider Java. Java is an object-oriented programming language with no language level support for real lexical closures. As a work-around Java programmers use anonymous inner classes that can close over the variables available in lexical scope (provided they're final). In this sense, objects are poor man's closures.
Closures are poor man's objects.
Consider Haskell. Haskell is a functional language with no language level support for real objects. However they can be modeled using closures, as described in this excellent paper by Oleg Kiselyov and Ralf Lammel. In this sense, closures are poor man's objects.
If you come from an OO background, you'll probably find thinking in terms of objects more natural, and may therefore think of them as a more fundamental concept than closures. If you come from a FP background, you might find thinking in terms of closures more natural, and may therefore think of them as a more fundamental concept than objects.
Moral of the story is that closures and objects are ideas that are expressible in terms of each other, and none is more fundamental than the other. That's all there is to the statement under consideration.
In philosophy, this is referred to as model dependent realism.
The point is that closures and objects accomplish the same goal: encapsulation of data and/or functionality in a single, logical unit.
For example, you might make a Python class that represents a dog like this:
class Dog(object):
def __init__(self):
self.breed = "Beagle"
self.height = 12
self.weight = 15
self.age = 1
def feed(self, amount):
self.weight += amount / 5.0
def grow(self):
self.weight += 2
self.height += .25
def bark(self):
print "Bark!"
And then I instantiate the class as an object
>>> Shaggy = Dog()
The Shaggy object has data and functionality built in. When I call Shaggy.feed(5), he gains a pound. That pound is stored in variable that's stored as an attribute of the object, which more or less means that it's in the objects internal scope.
If I was coding some Javascript, I'd do something similar:
var Shaggy = function() {
var breed = "Beagle";
var height = 12;
var weight = 15;
var age = 1;
return {
feed : function(){
weight += amount / 5.0;
},
grow : function(){
weight += 2;
height += .25;
},
bark : function(){
window.alert("Bark!");
},
stats : function(){
window.alert(breed "," height "," weight "," age);
}
}
}();
Here, instead of creating a scope within an object, I've created a scope within a function and then called that function. The function returns a JavaScript object composed of some functions. Because those functions access data that was allocated in the local scope, the memory isn't reclaimed, allowing you to continue to use them through the interface provided by the closure.
An object, at its simplest, is just a collection of state and functions that operate on that state. A closure is also a collection of state and a function that operates on that state.
Let's say I call a function that takes a callback. In this callback, I need to operate on some state known before the function call. I can create an object that embodies this state ("fields") and contains a member function ("method") that performs as the callback. Or, I could take the quick and easy ("poor man's") route and create a closure.
As an object:
class CallbackState{
object state;
public CallbackState(object state){this.state = state;}
public void Callback(){
// do something with state
}
}
void Foo(){
object state = GenerateState();
CallbackState callback = new CallbackState(state);
PerformOperation(callback.Callback);
}
This is pseudo-C#, but is similar in concept to other OO languages. As you can see, there's a fair amount of boilerplate involved with the callback class to manage the state. This would be much simpler using a closure:
void Foo(){
object state = GenerateState();
PerformOperation(()=>{/*do something with state*/});
}
This is a lambda (again, in C# syntax, but the concept is similar in other languages that support closures) that gives us all the capabilities of the class, without having to write, use, and maintain a separate class.
You'll also hear the corollary: "objects are a poor man's closure". If I can't or won't take advantage of closures, then I am forced to do their work using objects, as in my first example. Although objects provide more functionality, closures are often a better choice where a closure will work, for the reasons already stated.
Hence, a poor man without objects can often get the job done with closures, and a poor man without closures can get the job done using objects. A rich man has both and uses the right one for each job.
EDITED: The title of the question does not include "vice versa" so I'll try not to assume the asker's intent.
The two common camps are functional vs imperative languages. Both are tools that can accomplish similar tasks in different ways with different sets of concerns.
Closures are poor man's objects.
Objects are poor man's closures.
Individually, each statement usually means the author has a some bias, one way or another, usually rooted in their comfort with one language or class of language vs discomfort with another. If not bias, they may be constrained with one environment or the other. The authors I read that say this sort of thing are usually the zealot, purist or language religious types. I avoid the language religious types if possible.
Closures are poor man's objects. Objects are poor man's closures.
The author of that is a "pragmatist" and also pretty clever. It means the author appreciates both points of view and appreciates they are conceptually one and the same. This is my sort of fellow.
Just so much sugar, as closures hide anonymous objects under their skirts.
"Objects are a poor man's closures" isn't just a statement of some theoretical equivalence — it's a common Java idiom. It's very common to use anonymous classes to wrap up a function that captures the current state. Here's how it's used:
public void foo() {
final String message = "Hey ma, I'm closed over!";
SwingUtilities.invokeLater(new Runnable() {
public void run() {
System.out.println(message);
}
});
}
This even looks a lot like the equivalent code using a closure in another language. For example, using Objective-C blocks (since Objective-C is reasonably similar to Java):
void foo() {
NSString *message = #"Hey ma, I'm closed over!";
[[NSOperationQueue currentQueue] addOperationWithBlock:^{
printf("%s\n", [message UTF8String]);
}];
}
The only real difference is that the functionality is wrapped in the new Runnable() anonymous class instance in the Java version.
That objects can be used as a replacement for closures is quite easy to understand, you just place the captured state in the object and the calling as a method. Indeed for example C++ lambda closures are implemented as objects (things are sort of tricky for C++ because the language doesn't provide garbage collection and true closures with mutable shared state are therefore hard to implement correctly because of the lifetime of captured context).
The opposite (closures can be used as objects) is less observed but it's IMO a very powerful technique... consider for example (Python)
def P2d(x, y):
def f(cmd, *args):
nonlocal x, y
if cmd == "x": return x
if cmd == "y": return y
if cmd == "set-x": x = args[0]
if cmd == "set-y": y = args[0]
return f
The function P2d returns a closure that captured the two values of x and y. The closure then provide access for reading and writing to them using a command. For example
p = P2d(10, 20)
p("x") # --> 10
p("set-x", 99)
p("x") # --> 99
so the closure is behaving like an object; moreover as any access is going through the command interface it's very easy to implement delegation, inheritance, computed attributes etc.
The nice book "Let Over Lambda" builds over this idea using Lisp as a language, but any language that supports closures can use this technique (in Lisp the advantage is that you can also bend the syntax using macros and read macros to improve usability and automatically generate all boilerplate code). The title of the book is exactly about this... a let wrapping a lambda:
(defun p2d (x y)
(let ((x x) (y y))
(lambda (cmd &rest args)
(cond
((eq cmd 'x) x)
((eq cmd 'y) y)
((eq cmd 'set-x) (setq x (first args)))
((eq cmd 'set-y) (setq y (first args)))))))
Actually I'm not sure I agree with the "poor" adjective in this approach.
Locked. This question and its answers are locked because the question is off-topic but has historical significance. It is not currently accepting new answers or interactions.
One of the topics that seems to come up regularly on mailing lists and online discussions is the merits (or lack thereof) of doing a Computer Science Degree. An argument that seems to come up time and again for the negative party is that they have been coding for some number of years and they have never used recursion.
So the question is:
What is recursion?
When would I use recursion?
Why don't people use recursion?
There are a number of good explanations of recursion in this thread, this answer is about why you shouldn't use it in most languages.* In the majority of major imperative language implementations (i.e. every major implementation of C, C++, Basic, Python, Ruby,Java, and C#) iteration is vastly preferable to recursion.
To see why, walk through the steps that the above languages use to call a function:
space is carved out on the stack for the function's arguments and local variables
the function's arguments are copied into this new space
control jumps to the function
the function's code runs
the function's result is copied into a return value
the stack is rewound to its previous position
control jumps back to where the function was called
Doing all of these steps takes time, usually a little bit more than it takes to iterate through a loop. However, the real problem is in step #1. When many programs start, they allocate a single chunk of memory for their stack, and when they run out of that memory (often, but not always due to recursion), the program crashes due to a stack overflow.
So in these languages recursion is slower and it makes you vulnerable to crashing. There are still some arguments for using it though. In general, code written recursively is shorter and a bit more elegant, once you know how to read it.
There is a technique that language implementers can use called tail call optimization which can eliminate some classes of stack overflow. Put succinctly: if a function's return expression is simply the result of a function call, then you don't need to add a new level onto the stack, you can reuse the current one for the function being called. Regrettably, few imperative language-implementations have tail-call optimization built in.
* I love recursion. My favorite static language doesn't use loops at all, recursion is the only way to do something repeatedly. I just don't think that recursion is generally a good idea in languages that aren't tuned for it.
** By the way Mario, the typical name for your ArrangeString function is "join", and I'd be surprised if your language of choice doesn't already have an implementation of it.
Simple english example of recursion.
A child couldn't sleep, so her mother told her a story about a little frog,
who couldn't sleep, so the frog's mother told her a story about a little bear,
who couldn't sleep, so the bear's mother told her a story about a little weasel...
who fell asleep.
...and the little bear fell asleep;
...and the little frog fell asleep;
...and the child fell asleep.
In the most basic computer science sense, recursion is a function that calls itself. Say you have a linked list structure:
struct Node {
Node* next;
};
And you want to find out how long a linked list is you can do this with recursion:
int length(const Node* list) {
if (!list->next) {
return 1;
} else {
return 1 + length(list->next);
}
}
(This could of course be done with a for loop as well, but is useful as an illustration of the concept)
Whenever a function calls itself, creating a loop, then that's recursion. As with anything there are good uses and bad uses for recursion.
The most simple example is tail recursion where the very last line of the function is a call to itself:
int FloorByTen(int num)
{
if (num % 10 == 0)
return num;
else
return FloorByTen(num-1);
}
However, this is a lame, almost pointless example because it can easily be replaced by more efficient iteration. After all, recursion suffers from function call overhead, which in the example above could be substantial compared to the operation inside the function itself.
So the whole reason to do recursion rather than iteration should be to take advantage of the call stack to do some clever stuff. For example, if you call a function multiple times with different parameters inside the same loop then that's a way to accomplish branching. A classic example is the Sierpinski triangle.
You can draw one of those very simply with recursion, where the call stack branches in 3 directions:
private void BuildVertices(double x, double y, double len)
{
if (len > 0.002)
{
mesh.Positions.Add(new Point3D(x, y + len, -len));
mesh.Positions.Add(new Point3D(x - len, y - len, -len));
mesh.Positions.Add(new Point3D(x + len, y - len, -len));
len *= 0.5;
BuildVertices(x, y + len, len);
BuildVertices(x - len, y - len, len);
BuildVertices(x + len, y - len, len);
}
}
If you attempt to do the same thing with iteration I think you'll find it takes a lot more code to accomplish.
Other common use cases might include traversing hierarchies, e.g. website crawlers, directory comparisons, etc.
Conclusion
In practical terms, recursion makes the most sense whenever you need iterative branching.
Recursion is a method of solving problems based on the divide and conquer mentality.
The basic idea is that you take the original problem and divide it into smaller (more easily solved) instances of itself, solve those smaller instances (usually by using the same algorithm again) and then reassemble them into the final solution.
The canonical example is a routine to generate the Factorial of n. The Factorial of n is calculated by multiplying all of the numbers between 1 and n. An iterative solution in C# looks like this:
public int Fact(int n)
{
int fact = 1;
for( int i = 2; i <= n; i++)
{
fact = fact * i;
}
return fact;
}
There's nothing surprising about the iterative solution and it should make sense to anyone familiar with C#.
The recursive solution is found by recognising that the nth Factorial is n * Fact(n-1). Or to put it another way, if you know what a particular Factorial number is you can calculate the next one. Here is the recursive solution in C#:
public int FactRec(int n)
{
if( n < 2 )
{
return 1;
}
return n * FactRec( n - 1 );
}
The first part of this function is known as a Base Case (or sometimes Guard Clause) and is what prevents the algorithm from running forever. It just returns the value 1 whenever the function is called with a value of 1 or less. The second part is more interesting and is known as the Recursive Step. Here we call the same method with a slightly modified parameter (we decrement it by 1) and then multiply the result with our copy of n.
When first encountered this can be kind of confusing so it's instructive to examine how it works when run. Imagine that we call FactRec(5). We enter the routine, are not picked up by the base case and so we end up like this:
// In FactRec(5)
return 5 * FactRec( 5 - 1 );
// which is
return 5 * FactRec(4);
If we re-enter the method with the parameter 4 we are again not stopped by the guard clause and so we end up at:
// In FactRec(4)
return 4 * FactRec(3);
If we substitute this return value into the return value above we get
// In FactRec(5)
return 5 * (4 * FactRec(3));
This should give you a clue as to how the final solution is arrived at so we'll fast track and show each step on the way down:
return 5 * (4 * FactRec(3));
return 5 * (4 * (3 * FactRec(2)));
return 5 * (4 * (3 * (2 * FactRec(1))));
return 5 * (4 * (3 * (2 * (1))));
That final substitution happens when the base case is triggered. At this point we have a simple algrebraic formula to solve which equates directly to the definition of Factorials in the first place.
It's instructive to note that every call into the method results in either a base case being triggered or a call to the same method where the parameters are closer to a base case (often called a recursive call). If this is not the case then the method will run forever.
Recursion is solving a problem with a function that calls itself. A good example of this is a factorial function. Factorial is a math problem where factorial of 5, for example, is 5 * 4 * 3 * 2 * 1. This function solves this in C# for positive integers (not tested - there may be a bug).
public int Factorial(int n)
{
if (n <= 1)
return 1;
return n * Factorial(n - 1);
}
Recursion refers to a method which solves a problem by solving a smaller version of the problem and then using that result plus some other computation to formulate the answer to the original problem. Often times, in the process of solving the smaller version, the method will solve a yet smaller version of the problem, and so on, until it reaches a "base case" which is trivial to solve.
For instance, to calculate a factorial for the number X, one can represent it as X times the factorial of X-1. Thus, the method "recurses" to find the factorial of X-1, and then multiplies whatever it got by X to give a final answer. Of course, to find the factorial of X-1, it'll first calculate the factorial of X-2, and so on. The base case would be when X is 0 or 1, in which case it knows to return 1 since 0! = 1! = 1.
Consider an old, well known problem:
In mathematics, the greatest common divisor (gcd) … of two or more non-zero integers, is the largest positive integer that divides the numbers without a remainder.
The definition of gcd is surprisingly simple:
where mod is the modulo operator (that is, the remainder after integer division).
In English, this definition says the greatest common divisor of any number and zero is that number, and the greatest common divisor of two numbers m and n is the greatest common divisor of n and the remainder after dividing m by n.
If you'd like to know why this works, see the Wikipedia article on the Euclidean algorithm.
Let's compute gcd(10, 8) as an example. Each step is equal to the one just before it:
gcd(10, 8)
gcd(10, 10 mod 8)
gcd(8, 2)
gcd(8, 8 mod 2)
gcd(2, 0)
2
In the first step, 8 does not equal zero, so the second part of the definition applies. 10 mod 8 = 2 because 8 goes into 10 once with a remainder of 2. At step 3, the second part applies again, but this time 8 mod 2 = 0 because 2 divides 8 with no remainder. At step 5, the second argument is 0, so the answer is 2.
Did you notice that gcd appears on both the left and right sides of the equals sign? A mathematician would say this definition is recursive because the expression you're defining recurs inside its definition.
Recursive definitions tend to be elegant. For example, a recursive definition for the sum of a list is
sum l =
if empty(l)
return 0
else
return head(l) + sum(tail(l))
where head is the first element in a list and tail is the rest of the list. Note that sum recurs inside its definition at the end.
Maybe you'd prefer the maximum value in a list instead:
max l =
if empty(l)
error
elsif length(l) = 1
return head(l)
else
tailmax = max(tail(l))
if head(l) > tailmax
return head(l)
else
return tailmax
You might define multiplication of non-negative integers recursively to turn it into a series of additions:
a * b =
if b = 0
return 0
else
return a + (a * (b - 1))
If that bit about transforming multiplication into a series of additions doesn't make sense, try expanding a few simple examples to see how it works.
Merge sort has a lovely recursive definition:
sort(l) =
if empty(l) or length(l) = 1
return l
else
(left,right) = split l
return merge(sort(left), sort(right))
Recursive definitions are all around if you know what to look for. Notice how all of these definitions have very simple base cases, e.g., gcd(m, 0) = m. The recursive cases whittle away at the problem to get down to the easy answers.
With this understanding, you can now appreciate the other algorithms in Wikipedia's article on recursion!
A function that calls itself
When a function can be (easily) decomposed into a simple operation plus the same function on some smaller portion of the problem. I should say, rather, that this makes it a good candidate for recursion.
They do!
The canonical example is the factorial which looks like:
int fact(int a)
{
if(a==1)
return 1;
return a*fact(a-1);
}
In general, recursion isn't necessarily fast (function call overhead tends to be high because recursive functions tend to be small, see above) and can suffer from some problems (stack overflow anyone?). Some say they tend to be hard to get 'right' in non-trivial cases but I don't really buy into that. In some situations, recursion makes the most sense and is the most elegant and clear way to write a particular function. It should be noted that some languages favor recursive solutions and optimize them much more (LISP comes to mind).
A recursive function is one which calls itself. The most common reason I've found to use it is traversing a tree structure. For example, if I have a TreeView with checkboxes (think installation of a new program, "choose features to install" page), I might want a "check all" button which would be something like this (pseudocode):
function cmdCheckAllClick {
checkRecursively(TreeView1.RootNode);
}
function checkRecursively(Node n) {
n.Checked = True;
foreach ( n.Children as child ) {
checkRecursively(child);
}
}
So you can see that the checkRecursively first checks the node which it is passed, then calls itself for each of that node's children.
You do need to be a bit careful with recursion. If you get into an infinite recursive loop, you will get a Stack Overflow exception :)
I can't think of a reason why people shouldn't use it, when appropriate. It is useful in some circumstances, and not in others.
I think that because it's an interesting technique, some coders perhaps end up using it more often than they should, without real justification. This has given recursion a bad name in some circles.
Recursion is an expression directly or indirectly referencing itself.
Consider recursive acronyms as a simple example:
GNU stands for GNU's Not Unix
PHP stands for PHP: Hypertext Preprocessor
YAML stands for YAML Ain't Markup Language
WINE stands for Wine Is Not an Emulator
VISA stands for Visa International Service Association
More examples on Wikipedia
Recursion works best with what I like to call "fractal problems", where you're dealing with a big thing that's made of smaller versions of that big thing, each of which is an even smaller version of the big thing, and so on. If you ever have to traverse or search through something like a tree or nested identical structures, you've got a problem that might be a good candidate for recursion.
People avoid recursion for a number of reasons:
Most people (myself included) cut their programming teeth on procedural or object-oriented programming as opposed to functional programming. To such people, the iterative approach (typically using loops) feels more natural.
Those of us who cut our programming teeth on procedural or object-oriented programming have often been told to avoid recursion because it's error prone.
We're often told that recursion is slow. Calling and returning from a routine repeatedly involves a lot of stack pushing and popping, which is slower than looping. I think some languages handle this better than others, and those languages are most likely not those where the dominant paradigm is procedural or object-oriented.
For at least a couple of programming languages I've used, I remember hearing recommendations not to use recursion if it gets beyond a certain depth because its stack isn't that deep.
A recursive statement is one in which you define the process of what to do next as a combination of the inputs and what you have already done.
For example, take factorial:
factorial(6) = 6*5*4*3*2*1
But it's easy to see factorial(6) also is:
6 * factorial(5) = 6*(5*4*3*2*1).
So generally:
factorial(n) = n*factorial(n-1)
Of course, the tricky thing about recursion is that if you want to define things in terms of what you have already done, there needs to be some place to start.
In this example, we just make a special case by defining factorial(1) = 1.
Now we see it from the bottom up:
factorial(6) = 6*factorial(5)
= 6*5*factorial(4)
= 6*5*4*factorial(3) = 6*5*4*3*factorial(2) = 6*5*4*3*2*factorial(1) = 6*5*4*3*2*1
Since we defined factorial(1) = 1, we reach the "bottom".
Generally speaking, recursive procedures have two parts:
1) The recursive part, which defines some procedure in terms of new inputs combined with what you've "already done" via the same procedure. (i.e. factorial(n) = n*factorial(n-1))
2) A base part, which makes sure that the process doesn't repeat forever by giving it some place to start (i.e. factorial(1) = 1)
It can be a bit confusing to get your head around at first, but just look at a bunch of examples and it should all come together. If you want a much deeper understanding of the concept, study mathematical induction. Also, be aware that some languages optimize for recursive calls while others do not. It's pretty easy to make insanely slow recursive functions if you're not careful, but there are also techniques to make them performant in most cases.
Hope this helps...
I like this definition:
In recursion, a routine solves a small part of a problem itself, divides the problem into smaller pieces, and then calls itself to solve each of the smaller pieces.
I also like Steve McConnells discussion of recursion in Code Complete where he criticises the examples used in Computer Science books on Recursion.
Don't use recursion for factorials or Fibonacci numbers
One problem with
computer-science textbooks is that
they present silly examples of
recursion. The typical examples are
computing a factorial or computing a
Fibonacci sequence. Recursion is a
powerful tool, and it's really dumb to
use it in either of those cases. If a
programmer who worked for me used
recursion to compute a factorial, I'd
hire someone else.
I thought this was a very interesting point to raise and may be a reason why recursion is often misunderstood.
EDIT:
This was not a dig at Dav's answer - I had not seen that reply when I posted this
1.)
A method is recursive if it can call itself; either directly:
void f() {
... f() ...
}
or indirectly:
void f() {
... g() ...
}
void g() {
... f() ...
}
2.) When to use recursion
Q: Does using recursion usually make your code faster?
A: No.
Q: Does using recursion usually use less memory?
A: No.
Q: Then why use recursion?
A: It sometimes makes your code much simpler!
3.) People use recursion only when it is very complex to write iterative code. For example, tree traversal techniques like preorder, postorder can be made both iterative and recursive. But usually we use recursive because of its simplicity.
Here's a simple example: how many elements in a set. (there are better ways to count things, but this is a nice simple recursive example.)
First, we need two rules:
if the set is empty, the count of items in the set is zero (duh!).
if the set is not empty, the count is one plus the number of items in the set after one item is removed.
Suppose you have a set like this: [x x x]. let's count how many items there are.
the set is [x x x] which is not empty, so we apply rule 2. the number of items is one plus the number of items in [x x] (i.e. we removed an item).
the set is [x x], so we apply rule 2 again: one + number of items in [x].
the set is [x], which still matches rule 2: one + number of items in [].
Now the set is [], which matches rule 1: the count is zero!
Now that we know the answer in step 4 (0), we can solve step 3 (1 + 0)
Likewise, now that we know the answer in step 3 (1), we can solve step 2 (1 + 1)
And finally now that we know the answer in step 2 (2), we can solve step 1 (1 + 2) and get the count of items in [x x x], which is 3. Hooray!
We can represent this as:
count of [x x x] = 1 + count of [x x]
= 1 + (1 + count of [x])
= 1 + (1 + (1 + count of []))
= 1 + (1 + (1 + 0)))
= 1 + (1 + (1))
= 1 + (2)
= 3
When applying a recursive solution, you usually have at least 2 rules:
the basis, the simple case which states what happens when you have "used up" all of your data. This is usually some variation of "if you are out of data to process, your answer is X"
the recursive rule, which states what happens if you still have data. This is usually some kind of rule that says "do something to make your data set smaller, and reapply your rules to the smaller data set."
If we translate the above to pseudocode, we get:
numberOfItems(set)
if set is empty
return 0
else
remove 1 item from set
return 1 + numberOfItems(set)
There's a lot more useful examples (traversing a tree, for example) which I'm sure other people will cover.
Well, that's a pretty decent definition you have. And wikipedia has a good definition too. So I'll add another (probably worse) definition for you.
When people refer to "recursion", they're usually talking about a function they've written which calls itself repeatedly until it is done with its work. Recursion can be helpful when traversing hierarchies in data structures.
An example: A recursive definition of a staircase is:
A staircase consists of:
- a single step and a staircase (recursion)
- or only a single step (termination)
To recurse on a solved problem: do nothing, you're done.
To recurse on an open problem: do the next step, then recurse on the rest.
In plain English:
Assume you can do 3 things:
Take one apple
Write down tally marks
Count tally marks
You have a lot of apples in front of you on a table and you want to know how many apples there are.
start
Is the table empty?
yes: Count the tally marks and cheer like it's your birthday!
no: Take 1 apple and put it aside
Write down a tally mark
goto start
The process of repeating the same thing till you are done is called recursion.
I hope this is the "plain english" answer you are looking for!
A recursive function is a function that contains a call to itself. A recursive struct is a struct that contains an instance of itself. You can combine the two as a recursive class. The key part of a recursive item is that it contains an instance/call of itself.
Consider two mirrors facing each other. We've seen the neat infinity effect they make. Each reflection is an instance of a mirror, which is contained within another instance of a mirror, etc. The mirror containing a reflection of itself is recursion.
A binary search tree is a good programming example of recursion. The structure is recursive with each Node containing 2 instances of a Node. Functions to work on a binary search tree are also recursive.
This is an old question, but I want to add an answer from logistical point of view (i.e not from algorithm correctness point of view or performance point of view).
I use Java for work, and Java doesn't support nested function. As such, if I want to do recursion, I might have to define an external function (which exists only because my code bumps against Java's bureaucratic rule), or I might have to refactor the code altogether (which I really hate to do).
Thus, I often avoid recursion, and use stack operation instead, because recursion itself is essentially a stack operation.
You want to use it anytime you have a tree structure. It is very useful in reading XML.
Recursion as it applies to programming is basically calling a function from inside its own definition (inside itself), with different parameters so as to accomplish a task.
"If I have a hammer, make everything look like a nail."
Recursion is a problem-solving strategy for huge problems, where at every step just, "turn 2 small things into one bigger thing," each time with the same hammer.
Example
Suppose your desk is covered with a disorganized mess of 1024 papers. How do you make one neat, clean stack of papers from the mess, using recursion?
Divide: Spread all the sheets out, so you have just one sheet in each "stack".
Conquer:
Go around, putting each sheet on top of one other sheet. You now have stacks of 2.
Go around, putting each 2-stack on top of another 2-stack. You now have stacks of 4.
Go around, putting each 4-stack on top of another 4-stack. You now have stacks of 8.
... on and on ...
You now have one huge stack of 1024 sheets!
Notice that this is pretty intuitive, aside from counting everything (which isn't strictly necessary). You might not go all the way down to 1-sheet stacks, in reality, but you could and it would still work. The important part is the hammer: With your arms, you can always put one stack on top of the other to make a bigger stack, and it doesn't matter (within reason) how big either stack is.
Recursion is the process where a method call iself to be able to perform a certain task. It reduces redundency of code. Most recurssive functions or methods must have a condifiton to break the recussive call i.e. stop it from calling itself if a condition is met - this prevents the creating of an infinite loop. Not all functions are suited to be used recursively.
hey, sorry if my opinion agrees with someone, I'm just trying to explain recursion in plain english.
suppose you have three managers - Jack, John and Morgan.
Jack manages 2 programmers, John - 3, and Morgan - 5.
you are going to give every manager 300$ and want to know what would it cost.
The answer is obvious - but what if 2 of Morgan-s employees are also managers?
HERE comes the recursion.
you start from the top of the hierarchy. the summery cost is 0$.
you start with Jack,
Then check if he has any managers as employees. if you find any of them are, check if they have any managers as employees and so on. Add 300$ to the summery cost every time you find a manager.
when you are finished with Jack, go to John, his employees and then to Morgan.
You'll never know, how much cycles will you go before getting an answer, though you know how many managers you have and how many Budget can you spend.
Recursion is a tree, with branches and leaves, called parents and children respectively.
When you use a recursion algorithm, you more or less consciously are building a tree from the data.
In plain English, recursion means to repeat someting again and again.
In programming one example is of calling the function within itself .
Look on the following example of calculating factorial of a number:
public int fact(int n)
{
if (n==0) return 1;
else return n*fact(n-1)
}
Any algorithm exhibits structural recursion on a datatype if basically consists of a switch-statement with a case for each case of the datatype.
for example, when you are working on a type
tree = null
| leaf(value:integer)
| node(left: tree, right:tree)
a structural recursive algorithm would have the form
function computeSomething(x : tree) =
if x is null: base case
if x is leaf: do something with x.value
if x is node: do something with x.left,
do something with x.right,
combine the results
this is really the most obvious way to write any algorith that works on a data structure.
now, when you look at the integers (well, the natural numbers) as defined using the Peano axioms
integer = 0 | succ(integer)
you see that a structural recursive algorithm on integers looks like this
function computeSomething(x : integer) =
if x is 0 : base case
if x is succ(prev) : do something with prev
the too-well-known factorial function is about the most trivial example of
this form.
function call itself or use its own definition.